Watanabe shows that the free energy of asymptotically satisfies
I believe there's a mistake in what is being expressed here. The correct expression should be
where is the multiplicity. What is expressed in this post is the asymptotic expansion of what Watanabe calls the WBIC (which I think is slightly different to how the WBIC is defined in this post),
Asymptotically the WBIC and free energy agree up to the first two terms. In practice the WBIC is used as an approximation to the free energy (as the WBIC is easier to approximate via e.g MCMC methods).
I believe there's a mistake in what is being expressed here. The correct expression should be
where
Asymptotically the WBIC and free energy agree up to the first two terms. In practice the WBIC is used as an approximation to the free energy (as the WBIC is easier to approximate via e.g MCMC methods).